SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points...

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Main Authors: THOMAS A. HULSE, CHAN IEONG KUAN, DAVID LOWRY-DUDA, ALEXANDER WALKER
Format: Article
Language:English
Published: Cambridge University Press 2018-01-01
Series:Forum of Mathematics, Sigma
Subjects:
Online Access:https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article
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author THOMAS A. HULSE
CHAN IEONG KUAN
DAVID LOWRY-DUDA
ALEXANDER WALKER
author_facet THOMAS A. HULSE
CHAN IEONG KUAN
DAVID LOWRY-DUDA
ALEXANDER WALKER
author_sort THOMAS A. HULSE
collection DOAJ
description The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions $k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.
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spelling doaj.art-dd999b252e8f4d03997dfa9e10de1f822023-03-09T12:34:35ZengCambridge University PressForum of Mathematics, Sigma2050-50942018-01-01610.1017/fms.2018.26SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEMTHOMAS A. HULSE0CHAN IEONG KUAN1DAVID LOWRY-DUDA2ALEXANDER WALKER3Mathematics Department, Boston College, Chestnut Hill, MA 02467, USA;School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong Province, 519082, China;Warwick Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK;Department of Mathematics, Rutgers University Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA;The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions $k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article11N37 (primary)11F30 (secondary)
spellingShingle THOMAS A. HULSE
CHAN IEONG KUAN
DAVID LOWRY-DUDA
ALEXANDER WALKER
SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
Forum of Mathematics, Sigma
11N37 (primary)
11F30 (secondary)
title SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
title_full SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
title_fullStr SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
title_full_unstemmed SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
title_short SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM
title_sort second moments in the generalized gauss circle problem
topic 11N37 (primary)
11F30 (secondary)
url https://www.cambridge.org/core/product/identifier/S2050509418000269/type/journal_article
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AT chanieongkuan secondmomentsinthegeneralizedgausscircleproblem
AT davidlowryduda secondmomentsinthegeneralizedgausscircleproblem
AT alexanderwalker secondmomentsinthegeneralizedgausscircleproblem